Are you on two separate airlines? Depending on what the airline combo is, you might be able to get the bags tagged through to SLC. But no matter what you are going to have to pick up your bag at SEA. If it already has a tag on it for the next flight, you'll drop it off at a bag carousel. If it doesn't, you should approach the desk of the airline you're connecting to and they'll tag it in send it on its way.